Multisymplectic effectiveGeneral Boundary Field Theory

Jose A. Zapata

Centro de Ciencias Matematicas, UNAM, Mexico

zapata@matmor.unam.mx

September 2013

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What is presented, its origin and close relatives

Discrete spacetime multisymplectic GBFTGeometry of covariant 1st order filed theoryDecimation for scalar 1st order modelsThe action and its variationThe multisymplectic formulaSymmetries and conserved quantitiesThe canonical frameworkExamples

Gauge theories and modified BF theories

Regularization, coarse graining and continuum limit

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What is presented

^ “Multisymplectic Effective General Boundary Field Theory”M. Arjang and J. A. Zapata (to appear)

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A continuation of:

I J. A. Zapata (2004),–Loop quantization from a LGT perspective–

I E. Manrique, R. Oeckl, A. Weber and J. A. Zapata (2006),–Loop quantization as a continuum limit–

I M. P. Reisenberger (1997, 1994),–Classical discrete gauge theory and gravity / S F models–

I R. Oeckl (2003- ),–General Boundary Field Theory–

I A. P. Veselov (1988),–On discrete time hamiltonian systems–

I J. E. Marsden, G. W. Patrick, S. Shkoller (1998),–On discrete spacetime multisymplectic field theory–

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Closely related to:

I Gambini, Pullin, Di Bartolo, Porto (2002)–Consistent discretizations–

I Gambini, Pullin, Di Bartolo, Campiglia (2006)–Uniform discretizations–

I Dittrich, Bahr, Hoehn (2009)–Variational integrators and improved actions–

I Halvorsen, Sørensen, Christiansen (2011)–Noether’s theorem for spacetime simplicial gauge theory–

I Kogut, Susskind (1975)–Hamiltonian Formulation of Lattice Gauge Theories–

For regularization see

I Calcagni, Oriti, Thurigen (2013)–Structure for discrete Hodge star and Laplacian–

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Geometry of covariant 1st order filed theory

Continuum histories are local sectionsM ⊃ U

φ−→ E of a bundle with standard fibre F .Physical motions are selected by Hamilton’s principle with action

S(φ) =

∫UL(j1φ) , where j1φ(x) = (x , φ(x),Dφ(x)).

Its variation can be written as

dS(φ) · δφ = −∫U

(j1φ)∗(j1(δφ)yΩL) +

∫∂U

(j1φ)∗(j1(δφ)yΘL)

with ΘL, ΩL differential forms on J1Y .If we further define ΩL = −dΘL then

** φ solution, v ,w first var. =⇒∫∂U(j1φ)∗(vywyΩL) = 0 **

* S G-inv., ξ ∈ g gives vξ first var. =⇒∫∂U(j1φ)∗(vξyΘL) = 0 *

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Decimation for scalar 1st order models

Decimated local record of a history in 1st order format

νφ7−→ (ν, φν ∈ F , φτ ∈ Fτ⊂∂ν)

A variation δφ(ν) = v(ν) = (vν ∈ TφνF , vτ ∈ TφτFτ⊂∂ν)

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Scalar field’s action and its variation

Our starting point is a variational principle with action

S(φ) =∑ν∈Un

∆

L(φ(ν)) .

Its variation yields: (i) eqs. of motion and (ii) geometric structure

dS(φ)[v ] = −∑

U−∂Uφ∗(vyΩL) +

∑∂U

φ∗(vyΘL)

where

ΘL(·, φ(τν)) =∂L

∂φτ(φ(ν))dφτ ,

ΩL(·, φ(ν)) = −∂L∂φ

(φ(ν))dφν −∑

τ∈(∂ν)n−1

∂L

∂φτ(φ(ν))dφτ .

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The multisymplectic formula

We define

ΩL(v(ν), w(ν), φ(τν)) = −d(ΘL|φ(τν))(v(ν), w(ν)).

The geometrical structure arises when only solutions and firstvariations are considered. dS(φ)[v ] =

∑∂U φ

∗(vyΘL) implies0 = −ddS =

∑∂U φ

∗(ΩL). More precisely,∑∂U

φ∗(wyvyΩL) = 0

for any first variations v ,w of any solution φ.This is the multisymplectic formula.

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Its meaning in GBFT language

(Sols[U(∂U)] = ×iSols[U(Bi )], diag(ωL,1, ωL,m)) is not trivial.

Solutions in U correlate the spaces of the boundary components.

IU : Sols[U]→ Sols[U(∂U)] = ×iSols[U(Bi )]

The multisymplectic formula implies

I ∗Udiag(ωL,1, ..., ωL,m) = 0 .

*** In particular, U ≈ Σ× [0, 1] ⊂ M with ∂Σ = ∅ leads to ... ***

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Symmetries and conserved quantities

The Lie group G acts on F and on histories. In 1st order format

(gφ)(ν) = (ν, g(φν), g(φτ )τ⊂∂ν)

L(g φ(ν)) = L(φ(ν)) ∀ ν, φ, g =⇒ S and Sols[U] are G inv.

ξ ∈ g induces a first variation vξ of any solution φ. Then

0 = dS(φ)[vξ] =∑∂φ(U)

vξyΘL.

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The canonical framework for dimM = 1

L(q(ν)) = L(ν, q−, q, q+) provides momentum 1-forms

(q−,∂L

∂q−(q(ν)))dq− ∈ T ∗Qν− ,

(q+,∂L

∂q+(q(ν)))dq+ ∈ T ∗Qν+

that can be used to define “Legendre transformations”f −L : Qν− × Qν → T ∗Qν− and f +

L : Qν × Qν+ → T ∗Qν+ .The relation between the described lagrangian structure and(T ∗Q, θ = pdq) is simply

ΘL(·, q(ν)−) = −(f −L )∗θ , ΘL(·, q(ν)+) = (f +L )∗θ.

Dynamics arises as evolution in the form of canonicaltransformations with L as their generating function [Veselov].

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The canonical framework for dimM > 1

In covariant field theory, J1Y ∗ hosts canonical kinematics(Θ,Ω = −dΘ) and dynamics is placed over it.

This canonical str. could be pulled back to the discrete framework.

In a sense, this is what we do.Another way to see it is that we use

I predetermined codim 1 surfaces

I endowed with a notion of vertical variation(constructed from each face of the surface)

to extract a 1dim system and proceed as in the previous slide.

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Example: Particle in euclidean space with a potential

1. L(ν, q−ν , qν , q+ν ) = m

2 (qν−q−ν

a )2 + m2 (q

+ν −qνa )2 − V (qν)

2. dS(q)[v ] =∑

ν∈Udiscv [L](q(ν)) =

∑ν∈Udisc

∂L∂q (q(ν)) · vν

+∑

ν,ν+1∈Udisc

(∂L∂q+ (q(ν)) + ∂L

∂q− (q(ν + 1)))· v+ν

+[ ∂L∂q− (q(1)) · v−1 + ∂L

∂q+ (q(n)) · v+n ]

3.7→ΘL(·,q(ν)−)=mgAB(qν−q−ν )B

a dq−A, ΘL(·,q(ν)+), ΩL(·,q(ν))

4. For solutions and first variations:(i) ** Conserv. of symplectic structure (indep. of V ) **ΩL(·, ·, q(ν)−) =− d(ΘL|q(ν)−) = ΩL(·, ·, q(ν)+)(ii) ** Symmetries lead to conserved quantities **V = 0⇒ transl. symm. ⇒ 0 = dS(q)[vξ] = vξyΘL|∂q(U)

⇒ conserv. of p ⇒ q1 − q−1 = q+n − qn

5. L time indep. 6⇒ energy conserv.: most systems are chaotic

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Example: Veselov’s top Q = SO(n)

1. L(ν, g−, g , g+) = 12 Tr(g−1Ig−)a + 1

2 Tr((g+)−1Ig)a

2. Veselov: It has a complete set of first integrals in involution.

3. vξ(ν) = (ξ−ν g−ν , ξνgν , ξ

+ν g

+ν )

4. ΘL(vξ, g−ν )

.= vξ[L](g(ν))|ν− = −1

2 Tr(ξ−ν g−ν IgT

ν )a

5. ΩL(vξ, wη, g−ν ) = −1

2 Tr((ξνη−ν − ηνξ−ν )g−ν IgT

ν )a

6. Conserv. of the symplectic str. (q solution, vξ, wη first var.)−1

2 Tr((ξ1η−1 − η1ξ

−1 )q−1 IqT1 ) = −1

2 Tr((ξnη−n − ηnξ−n )q−n Iq

Tn )

7. Conserv. of ang. momentum (q solution)m(q) = 1

2 (q−ν IqTν − qν Iq

−Tν ) = 1

2 (qν Iq+Tν − q+

ν IqTν )

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Example: Non linear 2d waves (see Marsden et al)

In a hyper cubical discr. L =∑

c Lc with L++(φ(ν)) =

1

2

[(φ0+ − φν

h

)2

−(φ1+ − φν

k

)2]

+ N(φν)

hk , etc

Even with a non linear term,∑

∂U φ∗(ΩL) = 0.

Our framework yields ΘL,ΩL (and gluing eqs.) indep. of N, e.g.

ΩL(·, ·, φ1+ν ) = −2h

kdφν ∧ dφ1+.

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Decimation for gauge theories and modified BF theories

Reisenberger’s discretization:

Record of a history in 1st order format νA7−→ (ν, hlν , krν)

Our record of the curvature at ν is g∂s = h−1l ′ k

−1r ′ kr hlν

A variation in 1st order format v(ν) = (vl ∈ ThlGν , vr ∈ Tkr ν)

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Gauge field’s action and its variation

S(A) =∑ν∈Un

∆

L(A(ν)) =∑ν∈Un

∆

L(ν, g∂s)

dS(A)[v ] = −∑

U−∂UA∗(vyΩL) +

∑∂U

A∗(vyΘL).

The resulting geometric structure is, then,

ΘL(·, A(τν)) =∑r⊂τ

∂L

∂kr(A(ν))dkr ,

ΩL(·, A(ν)) = −∑l⊂ν

∂L

∂hl(A(ν))dhl −

∑r⊂ν

∂L

∂kr(A(ν))dkr .

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Internal gauge symmetries

Internal gauge transfs. are local symms. of the action. Then:

I For internal gauge transformations over interior points of U=⇒ Redundancies of equations of motion.

I For internal gauge transformations over points Cτ ∈ ∂U=⇒ Constraints on possible boundary data for solutions.

I For internal gauge transformations over points Cσ ∈ ∂U=⇒ The dynamical content of boundary data can be

captured by the reduced variables k−1r ′ kr1.

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Example: Lattice gauge theory without fermions

1. SEuc(A) = β∑

ν⊂U∑

s<ν [1− 1n Re Tr(g∂s)]

2. vξ(ν) = (hlξl ∈ ThlGν , ξrkr ∈ Tkr ν)

3. dS(A)[v ] = −∑

U−∂U A∗(vyΩL) +∑

∂U A∗(vyΘL) with

ΘL(vξ, A(τν))= vξL(A(ν))|τ =−βn

Re∑r⊂τ

Tr(h−1l ′ k−1

r ′ ξrkrhl)

ΩL(v , A(ν))= vξL(A(ν)) =−βn

Re∑l⊂ν

Tr(h−1l ′ k−1

r ′ krhlξl)

+−βn

Re∑r⊂∂ν

Tr(h−1l ′ k−1

r ′ ξrkrhl)

4. 0 = −βn Re

∑∂U

∑r⊂τTr(ξr (ηl − ηl ′)− ηr (ξl − ξl ′)g∂s)

where ξr = krhl ξrh−1l k−1

r

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Example: Reisenberger’s model

S(A, e, ψ) =∑ν⊂U

[∑s<ν

es i Tr(J ig∂s)− 1

60

∑s,s<ν

ψijν es ies jsgn(s, s)]

The equations of motion were written by Reisenberger.Notice that at each ν the boundary dof are purely connection dof.

** ΘL = ΘL,BF ** Consider vξ(ν) with vξr = ξkr ∈ TkrSU(2)

ΘL(vξ, æ(τν)) = vξL(æ(ν))|τ =∑r⊂τ

esr i Tr(J ih−1l ′ k−1

r ′ ξrkrhl) ,

Thus, the multisymplectic formula,∑

∂U æ∗ΩL = 0,is independent of the constraint term.

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Regularization

Decimated Histories∆i∆−→ Histories Continuum

e.g. selecting closest sols. to “the free theory” (if it makes sense)

S∆ = i∗∆S

Reisenberger’s discretization can be obtained by intersecting a‘prime’ simplicial decomposition with its dual.Having two intersecting discretizations allows for a notion ofHodge dual for cochains that sets up a rich structure forregularization; see [Calcagni et al].We have this structure plus the simple gluing already mentioned.

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Coarse graining

Hists. at finer scale ∆′ ≥ ∆ are coarse grained decimating further:

φ∆′π7−→ φ∆

Hists. and vars. in 1st order format are coarse grained likewise.

Notice the difference with directly working on T ∗Q or J1Y ∗.

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Coarse graining: Correction of a model at ∆

Coarse graning hists. and vars. from ∆′ > ∆ can be used toimport solutions and first variations, but we can do better ...

Measures in the space of histories are also coarse grained.(In QFT and SFT there is a straight forward interp. of coarse gr.)They can be corrected by coarse graining from finer scales.** A probabilistic perfect measure at scale ∆ is one thatagrees with all its corrections. **(i) ∆-Effective eqs. of motion are not exactly satisfied, and(ii) cons. laws are not exactly satisfied either. (See next slide.)

In dim(M) = 1 we have access to Hamilton’s function at any scale.In dim(M) > 1 knowledge of bdary. cond. is only partial;thus, we have no access to it.

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Continuum limit

Fix a history φ and a variation δφ from the continuum,and let the measuring (decimation) scale get finer

lim∆→M

dS∆(φ∆) · δφ∆

7→ Eqs. of motion and geo. str. may converge [Veselov, Marsdenet al].This also gives the error at scale ∆.

– Should we? –

Specify a solution φ in the continuum as a sequence φ∆of approx. solutions to effective eqs. of motion(∆) s.t.(i) φ∆ → φ, and (ii) dS∆ converges (see [Gambini et al]),=⇒ Symmetries are a limit of “approx. symmetries(∆)”

(discussion Dittrich, Rovelli).

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